Suppose that in the definition of an Euler cycle, we drop the seemingly superfluous requirement that the Euler cycle visit every vertex and require only that the cycle include every edge. Show that now the theorem is false. Draw a graph that illustrates why the theorem is false.
I guess I might think of it this way: we could not have odd degree vertices, and now we don't need to visit those odd degree vertices. But I guess I'm confused because it seems like we still do, because we still need to cover the edges that connect those vertices.